Deriving Spin-1/2 and Orbital Stability from Geometric First Principles

We derive two fundamental quantum phenomena—half-integer spin and orbital stability—from a single geometric principle: double saturation of quantum indeterminacy. By postulating that stable quantum states require simultaneous saturation of both spatial (∆x · ∆p = ℏ/2) and temporal (∆E · ∆t = ℏ/2) uncertainty relations, we explain why ground states persist indefinitely while excited states decay, and why fermions exhibit the characteristic 720ř phase return. The derivation follows from elementary geometry: double saturation yields λ = 4πr, and since 4π ≈ 12.566 falls between integers 12 and 13, quantized systems must oscil-late between discrete configurations—the dynamic pattern observed as spin-1/2. The half-integer quantum numbers that puzzled the founders of quantum mechanics are not mysterious axioms but the numerical signature of this forced geometric oscillation. The framework deliberately works on the physical manifold rather than in abstract Hilbert space, using only Heisenberg uncertainty and de Broglie relations from the 1920s. The goal is ontological clarification: explaining the geometric origin of established phenomena rather than forecasting new ones.